Be sure this exam has 12 pages including the cover
The University of British Columbia
MATH 308, Section 101, Instructor Tai-Peng Tsai
Final Exam – December 2011
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No notes nor calculators.
Rules Governing Formal Examinations:
1. Each candidate must be prepared to produce, upon request, a
UBCcard for identification;
problem
max
2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions;
1.
10
2.
10
3.
10
4. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action;
(a) Having at the place of writing any books, papers or
memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners;
(b) Speaking or communicating with other candidates;
(c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received;
4.
15
5.
10
6.
10
7.
10
8.
15
5. Candidates must not destroy or mutilate any examination material; must hand in all examination papers, and must not take
any examination material from the examination room without
permission of the invigilator; and
9.
10
total
100
3. No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination;
6. Candidates must follow any additional examination rules or
directions communicated by the instructor or invigilator.
score
December 2011
Math 308 Section 101 Final Exam
1. Suppose a conic has two focus points F (4, 0) and F 0 (−4, 0).
(5 pt)
(a) Find the equation of the conic if a directrix has the equation x = 9.
(5 pt)
(b) Find the equation of the conic if the conic has eccentricity e = 2.
Page 2 of 12
December 2011
Math 308 Section 101 Final Exam
Page 3 of 12
(5 pt) 2. (a) Find the equation of the tangent line to the parabola x = y 2 at the point (1, −1).
(5 pt)
(b) Classify the conic
x2 − 6xy + 9y 2 = 1.
December 2011
Math 308 Section 101 Final Exam
Page 4 of 12
3. The triangle 4ABC has vertices A(−1, 2), B(−3, −1) and C(3, 1), and the points P (1, 13 ),
Q(1, 23 ), and R(− 53 , 1) lie on BC, CA and AB, respectively.
(7 pt)
(a) Determine the ratios in which P , Q and R divide the sides of the triangle.
(3 pt)
(b) Determine whether or not the lines AP , BQ and CR are concurrent.
December 2011
Math 308 Section 101 Final Exam
Page 5 of 12
(5 pt) 4. (a) Find the Euclidean transformation which is the counterclockwise rotation about the
π
point (2, 2) by angle .
3
(5 pt)
0
4
1
to the points
and
,
(b) Find the affine transformation which maps the points
2
0
1
−2
5
1
, respectively.
and
,
1
−4
−1
December 2011
(5 pt)
Math 308 Section 101 Final Exam
(c) Find the Möbius transformation which maps the points 1, 2, i to 1, 2, 3.
Page 6 of 12
December 2011
Math 308 Section 101 Final Exam
Page 7 of 12
1
(5 pt) 5. (a) Find the image of the line 3x + y = 4 under the affine transformation t(x) = Ax +
−1
4 5
with A =
.
1 1
(5 pt)
(b) Find the image of the circle |z| = 1 under the Möbius transformation M (z) =
iz + 1
.
z+i
December 2011
Math 308 Section 101 Final Exam
Page 8 of 12
(10 pt) 6. Let two circles S1 and S2 intersect in A and B, and let the diameters of S1 and S2 through B
←→
cut S2 and S1 in C and D. Let S3 be the circle containing B, C, and D. Show that line AB
passes through the center of circle S3 .
December 2011
Math 308 Section 101 Final Exam
Page 9 of 12
(10 pt) 7. The circles C1 and C2 in an Apollonian family of circles have the segments [−11, −6] and
[0, 16], respectively, of the x-axis as diameters. Determine the point circles of the family.
December 2011
Math 308 Section 101 Final Exam
Page 10 of 12
(7 pt) 8. (a) Find the d-line that passes through the two d-points i/3 and −1/3, and sketch it.
(8 pt)
1
i
(b) Find the Euclidean radius of the non-Euclidean circle C = {z ∈ D : d( , z) = }.
3
3
Do not evaluate or simplify the solution. Note ln 2 ≈ 0.6931 and ln 3 ≈ 1.0986.
December 2011
Math 308 Section 101 Final Exam
Page 11 of 12
(10 pt) 9. Answer YES or NO to the following questions. Anything else will not earn any partial credits.
Answering both YES and NO earns 0.
(a) Is it true that an affine transformation preserves ratio of lengths on parallel lines?
(b) Is it true that, for any point P inside triangle 4ABC, there is an affine transformation t
such that t(P ) is outside of triangle 4t(A)t(B)t(C)?
(c) Is it true that an ellipse can be mapped onto a circle by an affine transformation?
(d) Is it true that an ellipse can be mapped onto a circle by an inversive transformation?
(e) Is it true that an inversion is a Möbius transformation?
December 2011
Math 308 Section 101 Final Exam
Page 12 of 12
Some Formulas
eccentricity
ellipse
0≤e<1
hyperbola
e>1
parabola
e=1
standard eq
x2 y 2
+ 2 =1
a2
b
2
x
y2
− 2 =1
2
a
b
y 2 = 4ax
focus
directrix
(±ae, 0)
x = ±a/e
(±ae, 0)
x = ±a/e
(a, 0)
x = −a
tangent at (x1 , y1 )
x 1 x y1 y
+ 2 =1
a2
b
x 1 x y1 y
− 2 =1
a2
b
y1 y = 2a(x + x1 )
p
Above b = a |1 − e2 |
1. Rotation about the origin by angle θ is t
x
cos θ − sin θ
x
=
, or t(z) = eiθ z.
y
sin θ cos θ
y
2. Reflection
the origin with angle θ/2 to the x-axis is
about the linepassing
x
cos θ sin θ
x
, or t(z) = eiθ z̄.
=
t
y
sin θ − cos θ
y
3. The affine transformation of R2 mapping [ 00 ], [ 10 ] and [ 01 ] to P = [ pp12 ], Q = [ qq12 ], and
R = [ rr12 ] is
q − p1 r1 − p1
t(~x) = P~ + A~x, A = (P~Q|P~R) = 1
.
q2 − p2 r2 − p2
4. Affine transformations preserve collinearity, coincidence, parallel lines, and signed ratio of
lengths along the same direction.
AR BP CQ
5. Ceva’s and Menelaus’ theorems: P ∈ BC, Q ∈ CA, R ∈ AB, RB
· P C · QA = ±1 implies
coincidence or collinearity.
p
6. Inversion in the unit circle: (x, y) 7→ ( rx2 , ry2 ) where r = x2 + y 2 . Inversion in the circle
r2
|z − m| = r is z 7→ z̄−
m̄ + m.
7. The stereographic projection π : S2 → C satisfies π : (X, Y, Z) 7→
π −1 : x + iy 7→ x2 +y1 2 +1 (2x, 2y, x2 + y 2 − 1).
X+iY
1−Z
with
8. Inversive transformations on Ĉ are of the form t(z) = M (z) or t(z) = M (z̄) with
M (z) = az+b
cz+d , ad 6= bc. Isometries correspond to M (z) = az + b with |a| = 1.
9. Non-Euclidean transformations on D are t(z) = M (z) or t(z) = M (z̄) with M (z) =
|b| < |a|. An inversion in the d-line |z − α| = r with |α|2 = r2 + 1 is t(z) = αz̄−1
z̄−ᾱ .
az+b
,
b̄z+ā
|z1 −z2 |
10. The distance function on D is d(z1 , z2 ) = tanh−1 ( |1−z̄
) and d(z, 0) = tanh−1 (|z|). Note
1 z2 |
y = tanh x =
ex −e−x
ex +e−x
and x = tanh−1 y =
1
2
ln 1+y
1−y for 0 ≤ y < 1.